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Proof step 119 from proof int2bits-rec-2pot-neg-02 (previous step)

j % 2 ^ i < 0, true ∈ uint2bits(~(j / 2 ^ i)), false ∈ uint2bits(~(j / 2 ^ i)), true ∈ fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest, 0 ≤ i, j < 0, j + 2 ^ i < i, j + 2 ^ i ≤ 0, 0 < i, j / 2 ^ i ≤ 0, j / 2 ^ i < i ⊦ bits2int(bitsinvert(uint2bits(~(j / 2 ^ i)))) = bits2int(sgn(bitsinvert(uint2bits(1 - j / 2 ^ i)) +1)), sgn(bitsinvert(uint2bits(~(j / 2 ^ i)))) + bitsinvert(fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest) +1 = sgn(bitsinvert(uint2bits(1 - j / 2 ^ i)) +1) + fillsgn(i→n(i) + 1, sgn(bitsinvert(uint2bits(~(j % 2 ^ i))) +1)).rest, j % 2 ^ i = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 120:
⊦ 0 < i → sgn(bitsinvert(uint2bits(i)) +1) = int2bits(~ i)
Proof step 121:
0 < 1 - j / 2 ^ i, j % 2 ^ i < 0, true ∈ uint2bits(~(j / 2 ^ i)), false ∈ uint2bits(~(j / 2 ^ i)), true ∈ fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest, 0 ≤ i, j < 0, j + 2 ^ i < i, j + 2 ^ i ≤ 0, 0 < i, j / 2 ^ i ≤ 0, j / 2 ^ i < i ⊦ bits2int(bitsinvert(uint2bits(~(j / 2 ^ i)))) = bits2int(int2bits(~(1 - j / 2 ^ i))), sgn(bitsinvert(uint2bits(~(j / 2 ^ i)))) + bitsinvert(fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest) +1 = int2bits(~(1 - j / 2 ^ i)) + fillsgn(i→n(i) + 1, sgn(bitsinvert(uint2bits(~(j % 2 ^ i))) +1)).rest, j % 2 ^ i = 0

Rule argument

name: int2bits-neg-01 rotate: F subst: [i] → [1 - j / 2 ^ i]

Additional information

Used simplifier rules: bits2int-sgn: ⊦ bits2int(sgn(bits)) = bits2int(bits);% ⊦ i < j - i0 ↔ i + i0 < j;% ⊦ i + 0 = i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i < 1 ↔ i ≤ 0;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i ≤ j → (¬ ¬ j < i ↔ false); Used terms: 1 - j / 2 ^ i